# Difference between revisions of "Brahmagupta's Formula"

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<cmath>16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2)</cmath> | <cmath>16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2)</cmath> | ||

<cmath>16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)</cmath> | <cmath>16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)</cmath> | ||

− | <cmath>16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b | + | <cmath>16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-a+b)</cmath> |

<cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath> | <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath> | ||

<cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> |

## Latest revision as of 14:23, 30 March 2018

**Brahmagupta's Formula** is a formula for determining the area of a cyclic quadrilateral given only the four side lengths.

## Definition

Given a cyclic quadrilateral with side lengths , , , , the area can be found as:

where is the semiperimeter of the quadrilateral.

### Proofs

If we draw , we find that . Since , . Hence, . Multiplying by 2 and squaring, we get: Substituting results in By the Law of Cosines, . , so a little rearranging gives

## Similar formulas

Bretschneider's formula gives a formula for the area of a non-cyclic quadrilateral given only the side lengths; applying Ptolemy's Theorem to Bretschneider's formula reduces it to Brahmagupta's formula.

Brahmagupta's formula reduces to Heron's formula by setting the side length .

A similar formula which Brahmagupta derived for the area of a general quadrilateral is where is the semiperimeter of the quadrilateral. What happens when the quadrilateral is cyclic?

## Problems

### Intermediate

- is a cyclic quadrilateral that has an inscribed circle. The diagonals of intersect at . If and then the area of the inscribed circle of can be expressed as , where and are relatively prime positive integers. Determine . (Source)